Key slides on production, supply, etc.
Key elements to focus on in lecture (handout covered on your own)
Production function: returns to scale; how to compute it, marginal product of each input, how to compute it
Cost-minimisation problem, cost-minimising input choice: for interior solution isocost and isoquant are parallel; MRTS = input price ratio (see ‘conditional input demand’)
The ‘cost function’ and the total cost curve… with IRS, DRS… (Total costs, marginal costs, average costs, fixed costs: on your own)
Firm’s profit-maximisation: An unconstrained problem… optimal q where MR(q)=mc(q) if TR>TC; inverse-elasticity rule; Shut-down decision
(On your own: what is ‘Economic profit?’)
Perfect competition: Definition; Entry and exit of firms yield zero profit in the LR; ‘Price-taker’ firms (what leads to this, what it means); firm’s supply curve under perfect competition; Market supply curve under PC, short and LR; Welfare consequences of perfect competition (CS)
Also: reminder about estimating a supply curve
The Firm’s Production function
The Firm’s Production function
\[q = f( K, L, M, ...)\]
\[q = f(K, L)\]
e.g., ‘Cobb douglas with particular proportions and constant returns to scale’
\[q =2L^{1/3}K^{2/3}\]
or ‘Leonteif’
\[y=min\{a_1 x_1, a_2 x_2, ...a_n x_n \}\] . . .
Or ‘two-input Constant Elasticity of Substitution’:
\[x_1^\rho + x_2^\rho)^(1/\rho)\]
Marginal product of each input
Additional output from adding +1 unit of an input, holding other inputs constant.
Marginal Product of Labour—\(MP_L\): Slope of production function in units of labour (holding capital etc constant)
\[MP_L=\frac{\partial}{\partial L}f(K,L)\]
E.g., for \(q = f(K,L) = 2L^{1/3}K^{2/3}\) … \(MP_L(K,L) =\) ?
Check, does this production function have ‘diminishing marginal product of labour’?
Marginal rate of technical substitution
‘Rate you can trade off input for another holding production constant’
\[RTS(K,L) = MP_L/MP_K\]
E.g., for \(q = f(K,L) = L^{1/3}K^{2/3}\) … \(MP_L(K,L) =\frac{2}{3} L^{-\frac{2}{3}} K^\frac{2}{3}\) \(MP_K(K,L) =\frac{4}{3} L^{\frac{1}{3}} K{^-\frac{1}{3}}\)
\(MP_L/MP_K = \frac{1}{2} K/L\)
See also ‘elasticity of substitution’
Returns to scale
The rate at which output increases in response to a proportional increase in all inputs.
\[f(\mathbf{tx}) \text{ versus } tf(\mathbf{x})\]
Computing… If you know the production function, how do you know if the ‘returns to scale’ are increasing or decreasing?
Slide in a constant \(\alpha>1\) next to each input, simplify, and compare to the original production
E.g.:
\[Q(L,K) = L^{1/4}K^{1/2}\]
\[Q(\alpha L, \alpha K) = (\alpha L)^{1/4}(\alpha K)^{1/2}\] \[=\alpha^{1/4}\alpha^{1/2}L^{1/4}K^{1/2}=\alpha^{3/4}L^{1/4}K^{1/2}=\alpha^{3/4}Q(L,K)\]
\(\rightarrow\) So if we increase inputs by \(\alpha\) here, we increase output by \(\alpha^{3/4}<\alpha\), so DRS everywhere for this production function.
Isoquant maps, rate of technical substitution (RTS)
Cost minimizing input choice
Cost minimizing input choice
Formally a cost function’:
\[c(\mathbf{w},y) \equiv \min_{\mathbf{x}\in R^n_+}(\mathbf{w}\cdot \mathbf{x}) \text{ s.t. } f(\mathbf{x}) \geq y \]
For input price vector \(\mathbf{w>>0}\) and output \(y\).
Lagrangian optimisation w interior solution \(\mathbf{x}^*>>0\) \(\rightarrow\) there is some \(\lambda^*\in R\) s.t.
\(w_i = \lambda \frac{\partial f(\mathbf{x}^*)}{\partial x_i} \forall i\)
\(\rightarrow \frac{\partial f(\mathbf{x}^*)}{\partial x_i}/\frac{\partial f(\mathbf{x}^*)}{\partial x_j} = \frac{w_i}{w_j} \forall i,j\)
I.e., where input mix optimal, each input that is used yields same marginal product per £
Consider a combo of K and L, and the output this yields.
From here, firm can ‘substitute capital for labour’ at rate \(RTS(K,L)= MP_L/MP_K\) (& hold production constant)
If firm uses both K & L and chooses optimally \(\rightarrow\)
Set \(RTS(K,L)\) equal to the input price ratio of these inputs (\(w/r\))
\(\rightarrow\) ‘Same bang for the buck at optimal choices \(K^*\) and \(L^*\)’ i.e.,
\[\frac{MP_K(K^*,L^*)}{r}= \frac{MP_L(K^*,L^*)}{w}\]
Isocost and isoquant
Aside, for intuition and story-telling
If markets for labour and capital are (perfectly) competitive prices of inputs & outputs adjust so that:
‘bang for the buck’ (in revenue) is the same for all inputs and for all production processes
Inputs (workers, owners of capital) in every industry will be paid based on their (marginal) productivity …
Advanced question
Back to our ‘replaced by AI’ motivation, a simple model. Suppose:
One product in the economy with ‘constant returns to scale Cobb-Douglas production function:’
\[q = L^{a}K^{1-a}\]
This implies ‘optimising firms will spend a share \(\alpha\) on labour’.
Summing up
Summing up: Optimisation (given a production function and input prices)…
yields a (minimum) cost for every output \(q\) a firm chooses to produce.
Returns to scale
Returns to scale
Are bigger firms always more efficient? Do things get cheaper to produce the more we produce?
- Returns to scale
- The rate at which output increases in response to a proportional increase in all inputs.
- Constant returns to scale (CRS)
- If inputs increase by a factor of X, output increases by a factor equal to X.
E.g., doubling all inputs (labor, capital, land, etc) means exactly doubling all outputs
- Increasing returns to scale (IRS)
- If inputs increase by a factor of X, output increases by a factor greater than X.
- Decreasing returns to scale (DRS)
- If inputs increase by a factor of X, output increases by a factor less than X.
Arguments/reasons for scale (dis)economies
IRS
Fixed costs (incorporation, buildings, management, planning, R&D) spread over more units
Should always be able to at least ‘double everything’ and produce twice as much? (so at least CRS)
Scale allows specialisation
Arguments for DRS
- Limited resources in (relevant) economy; costs begin to rise
- Managerial issues and coordination problems, bigger ‘centre’ to lobby for favours. See ‘theories of the firm’
- Harder to give incentives to top manager/CEO?
Costs
Types of costs (‘Basic cost concepts’)
Fixed costs (FC): Costs that must be regularly incurred to remain in business (i.e., for any level of output), but that do not vary with the level of output
Variable costs (VC): Costs that increase with the quantity produced.
Sunk costs: Costs that have been incurred in the past that can never be recovered.
Sunk costs should not enter into any economic decisions.
FC from previous years are sunk costs; FC for future years are not.
Again: Economic profits and cost minimisation; 2-input model
Again: Economic profits and cost minimisation
2-input model
- Labour costs - wage rates \(w\)
- Capital costs - rental rate \(v\)
Total costs \(= TC = wL + vK\)
Economic profit = \(\pi\) = Total Revenues - Total Costs
\[\pi = Pq-wL-vK = P\times f(K,L)-wL-vK\]
(P: price of good)
Cost minimizing input choice, expansion path, ratio condition
Choose point where RTS = ratio of input prices
- RTS = (wage rate/rental rate) = w/v
- Same ratio of ‘marginal productivity/price’ for each input used
Which point on this curve will minimize cost? . . . The one on the lowest ‘isocost line’ … similar to budget line for consumers
Intuition
\[RTS = MP_L/MP_K = w/v\]
Cost-minimisation for each level of output \(\rightarrow\) firm’s expansion path
- Total cost curve: from cost of inputs along expansion path
Average and marginal cost curves
Read at home:
Average and marginal cost curves
… with differing returns to scale: Increasing, decreasing, constant, optimal scale
Skip: “Short and long run” costs for the firm
- Average cost
- Cost per unit of output
\[AC=TC/q\]
- TC = ‘summed’ marginal costs and fixed costs
- AC: ‘average marginal cost’ + FC/q
- Marginal cost
- Incremental cost of last unit produced
- I.e., additional cost of producing one more unit of output
- MC: Slope of (optimising) TC curve at a point
Shape of marginal cost curve depends on production function
Constant returns to scale: constant MC (and no FC)
Decreasing returns to scale: increasing MC
Increasing returns to scale: decreasing MC (and/or constant FC)
On estimating cost/production functions
Want to estimate product, firm, and industry production functions
Government (regulators etc), forecasters, strategists
Anti-trust, regulating natural monopolies, macro-growth issues and aggregate production, importance of human-K, impact of trade deals (winners/losers), impact of min. wage and labour laws, business strategy and competition, reacting to anticipated market changes…
Production functions \(\Leftrightarrow\) cost functions
- Amount that can be produced with any set of inputs \(\Leftrightarrow\) (Minimum) cost of producing any output
- Difficult to estimate these: Lack of publicly available data, lack of exogenous variation in input choices